The locus of complex number z satisfying arg[z−(1+i)]=⎧⎪
⎪⎨⎪
⎪⎩3π4;|z|≤|z−2|−π4;|z|>|z−2| is
A
a straight line passing through (2,0).
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B
a straight line passing through (2,0),(1,1).
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C
is a line segment.
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D
is a set of two rays.
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Solution
The correct option is D is a set of two rays. Given equation is arg[z−(1+i)]=⎧⎪
⎪⎨⎪
⎪⎩3π4;|z|≤|z−2|−π4;|z|>|z−2|
Let z=x+iy
When |z|≤|z−2| ⇒x2+y2≤(x−2)2+y2⇒0≤−4x+4⇒x≤1
When |z|>|z−2| ⇒x>1